Page 268 - Intelligent Digital Oil And Gas Fields

218 Intelligent Digital Oil and Gas Fields

where vector u has a dimension of m 1(m¼discretization of the reservoir
parameter, that is, the total number of representative grid cells) and the col-
umn vector v is the n t -length spectrum of transform coefficients. The n t col-
umns of the transform basis Φ represent the discrete basis functions with
length of m. The main objective of parameterization is to reduce the param-
eter dimension, that is, the dimension of vector v, with a compact/truncated
representation of Φ that contains only a few basis functions that are still able
to capture relevant model spatial information. Schematically, the parameter-
ization by linear transformation mapping is presented in Fig. 6.6, while
Table 6.2 lists the prevalent subspace model parameterization methods

x 3 f=F(x) f 3

f
x
x 2
−1
x=F (f) f 2
x 1
Fig. 6.6 Schematic representation of mapping from the parameter space to the feature
domain.

Table 6.2 Selected (Meta)heuristic Optimization Methods With Main Applications
Parameterization
Technique Reference
Singular value Yanai et al. (2011)
decomposition (SVD)
Karhunen-Loeve Newman (1996) and Jafarpour and McLaughlin (2007)
transform (KLT)
Fourier-space filter Maucec et al. (2007)
expansion
Principal component Honorio et al. (2015), Kang et al. (2015), and Chen et al.
analysis (PCA) (2014)
Discrete cosine transform Jafarpour and McLaughlin (2007, 2009) and Maucec
(DCT) et al. (2011, 2013a,b)
Grid-connectivity Bhark et al. (2011), Kang et al. (2014), and Kam et al.
transform (GCT) (2016)
Multidimensional scaling Scheidt and Caers (2009), Maucec et al. (2011, 2013a,b),
(MDS) and Arnold et al. (2016)
Note: Parameterization techniques SVD, KLT, and PCA are occasionally commonly referred to as proper
orthogonal decomposition (POD) techniques.